|
Date
|
Speaker
|
Title
|
|
September
11
|
Jingjun
Han
Johns Hopkins
|
Birational
boundedness of rationally connected Calabi--Yau 3-folds
We will show that rationally connected klt Calabi¡ªYau 3-folds form a
birationally bounded family, and such 3-folds with mld bounding away from 1
are bounded modulo flops. This is a joint work with Weichung Chen, Gabriele
Di Cerbo, Chen Jiang and Roberto Svaldi.
|
|
September
18
|
Zili
Zhang
Michigan
|
P=W:
a strange identity for affine Dynkin diagrams
Start with a compact Riemann surface X with marked points and a complex
reductive group G. According to Hitchin-Simpson's nonabelian Hodge theory,
the pair (X,G) comes with two new complex varieties: the character variety
M_B and the Higgs moduli M_D. I will present some aspects of this story and
discuss a new identity P=W indexed by affine Dynkin diagrams - occurring in
the singular cohomology groups of M_D and M_B, where P and W dwell. Based on
joint work with Junliang Shen.
|
|
September
25
|
Ziquan
Zhuang
Princeton
|
Birational
superrigidity and K-stability
Birational superrigidity and K-stability are properties of Fano varieties
that have many interesting geometric implications. For instance, birational
superrigidity implies non-rationality and K-stability is related to the
existence of Kähler-Einstein metrics. Nonetheless, both properties are hard
to verify in general. In this talk, I will first explain how to relate
birational superrigidity to K-stability using alpha invariants; I will then
outline a method of proving birational superrigidity that works quite well
with most families of index one Fano complete intersections and thereby also
verify their K-stability. This is partly based on joint work with Charlie
Stibitz and Yuchen Liu.
|
|
October
2
|
Shokurov
Johns Hopkins
|
Boundedness
and existence of n-complements
This talk is about the theorem of boundedness and existence of n-complements
of a local relative pair with boundar. The morphism of pair is supposed to be
an FT contraction. The local property means that the morphism is defined over
a neighborhood of (not necessarily closed) point. For the existence of
n-complements it is sufficient the existence of numerical or R-complements.
The boundedness means that for n-comlements of pairs of a fixed dimension it
is sufficient a finite set of positive integers n. Moreover, such sets has
some additional properties: divisibility and aproximation properties for
irrational numbers and vectors. The latter properties implies important
applications to certain questions and results about acc of some well-known
invariants of log pairs. E.g., acc of the log canonical thresholds. This
allows to give a new more simple proof for the finite generatedness of log
canonical ring and for the existence of flips.
|
|
October
9
|
Akash
Sengupta
Princeton
|
Fujita
invariant and the thin set in Manin's conjecture
Let X be a Fano variety defined over a number field with an associated height
function. Manin's conjecture predicts that, after removing a thin set, the
growth of the number of rational points of bounded height on X is governed by
certain geometric invariants (e.g. Fujita invariant of X). I'll talk about
how to use birational geometric methods to study the behavior of these
invariants and propose a description of the thin set in Manin's conjecture.
Part of this is joint work with Brian Lehmann and Sho Tanimoto.
|
|
October
16
|
Jihao
Liu
Utah
|
ACC
for MLDs and ACC for a-LCTs
Minimal log discrepancies and a-log canonical thresholds are important
invariants of singularities that play a fundamental role in higher
dimensional birational geometry. It is conjectured that both minimal log
discrepancies and a-log canonical thresholds satisfy the ACC property. In
this talk I will discuss the relationship between these two conjectures. In
particular, I will introduce my recent work which shows that these two
conjectures are equivalent for non-terminal singularities.
|
|
October
23
|
Oscar
Kivinen
UC Davis
|
Hecke
correspondences on Hilbert schemes of singular plane curves and knot homology
Let C be a locally planar complex curve with m irreducible components. In this
talk, I will define a certain subalgebra of the Weyl algebra on C^{2m} and
show how it acts on the homologies of the Hilbert schemes of points on C via
Hecke-like correspondences. I will then compare this action to similar
actions on the equivariant homologies of affine Springer fibers in type A
(i.e. local compactified Jacobians), and time permitting, certain knot
homologies of the links of the singularities.
|
|
October
30
|
Jakub
Witaszek
IAS
|
On
the Minimal Model Program in low characteristics
Despite the substantial progress on the Minimal Model Program for
three-dimensional varieties in characteristic p>5, so far little was known
for p no larger than 5. The goal of this talk is to shed some light on the
geometry of threefolds in low characteristic. This is based on joint work
with Christopher Hacon.
|
|
November
6
|
John
Lesieutre
Penn State
|
Some
remarks on arithmetic degrees
Suppose that f : X -> X is a rational self-map of an algebraic variety,
with everything defined over Qbar. If P is a Qbar-point of X, then the
arithmetic degree alpha_f(P) is a measure of the rate of growth of the
heights the points of f^n(P), and is a sort of "arithmetic entropy"
of f. Kawaguchi and Silverman made several conjectures about the properties
of alpha_f(P), including one relating it to the first dynamical degree of f,
itself a close relative of the topological entropy.
|
|
November
13
|
Stefano
Filipazzi
Utah
|
A
generalized canonical bundle formula and applications
Birkar and Zhang recently introduced the notion of generalized pair. These
pairs are closely related to the canonical bundle formula and have been a
fruitful tool for recent developments in birational geometry. In this talk, I
will introduce a version of the canonical bundle formula for generalized
pairs. This machinery allows us to develop a theory of adjunction and
inversion thereof for generalized pairs. I will conclude by discussing some
applications to a conjecture of Prokhorov and Shokurov.
|
|
November
27
|
Chuanhao
Wei
Stony Brook
|
Zeros
of log-one-forms and families of log-varieties
I will introduce the result about the relation between the zeros of
holomorphic log-one-forms and the log-Kodaira dimension, which is a natural
generalization of Popa and Schnell's result on zeros of one-forms. Some
geometric corollaries will be stated, e.g. algebraic Brody hyperbolicity of
log-smooth family of log-pairs of log-general type, which also serves as a
motivation to the main theorem.
|